Something like these http://www.amazon.com/Wiremold-C210-White-Cordmate-Kit/dp/B00004W3ZQ/ref=sr_1_2?ie=UTF8&qid=1354040365&sr=8-2&keywords=cable+hiding could help route cables around the right-hand edge of your built-in. You might need some cable extensions in order to do it cleanly, but these kinds of moldings will probably be your best bet.

What about mounting the stuff to the back of the TV? looks like it's not attached to the wall, so you can make a bracket and use the mounting screws. Might make the tv a little top heavy, but you could strap it to the wall or something with little wall anchors.

perhaps the Plug Hub might help:
http://www.apartmenttherapy.com/10-tech-gift-ideas-for-unorganized-friends-and-family-180497

I would move all electronics to the shelf that the tv rests on, then tie up any slack in the wires using twist ties, and try to hide the bundled wires behind the devices themselves. Then, if the row of wires coming out of them to the tv is still distracting or distasteful, use the solution such as the first one proposed here to hide them (that white plastic bar thingie.)

Look into a wireless IR repeater system. These allow you to put the component pretty much anywhere within about 100-200 feet of the TV, even hide them in a closet if you want, or put doors on your existing shelves. You might still need a single cable somewhere but it would be neater than what you have.

Adapting flippedcracker's idea, you could attach the boxes to the wall behind the TV, which would make the cable problem disappear.

I recently solved this problem at my house which has a similar layout. First I built a TV stand that completely covered the non-functional fireplace. It was essentially a 20" high box with a back cover that extended up to the mantle. It makes viewing much easier on the neck. Then I added a shelf to the top of the bookcases to make it deeper and thats where the cable box, DVD etc went. Some of the cords can slide behind the TV stand. There was still a nest of wires that had to reside on one of the shelves so I bought some $1 books cut the spines off and created a fake book cover to hide the power strip. I wish I could post a photo of the before and after for a better idea. Good luck! Its not an easy problem to solve.

Easy solutions that cost little to no money: Move all the components to one shelf (I'd recommend some type of shelving sytem to prevent stacking them all on top of each other) and bundle the cables together. Find white cabes to go to the tv (or paint the cable) to make it less of an eyesore.

If you have considered building a home theater, now might be the time. Many receivers (the central part of a home theater) allow you to connect all your devices into it and run a single cable (usually HDMI) to your tv/projector. This would allow you to move your components elsewhere and just have the single cable to worry about.

Ditto on Xarcady's comment on a wireless IR repeater system. A few years ago I purchased a Harmony Remote (amazing) and the model I purchased (the 890 I believe) came with it. It allows me to have my components in a closet and still have full functionality.

Buy or build a 6 inch high box to place on the top shelf, and rest the TV on top and put the cable boxes underneath.

I'd double check with the landlord about drilling holes. Since the shelving is painted it's easy to repair (50 cent cork in the hole , flexible saw to cut in flat. and a little white paint)

Even show him the idea and let the landlord drill if preferred.

I would drill one to get power to the top of the mantel. Then move all the components up there, use a wire wrap like most people are suggesting. Then you don't have to get the TV cable down to the components.. everything is on one plain where you can wrangle all the wires. The messiness comes from crossing shelves more than anything.

I agree that putting the components all up on the same level, then tucking the wires neatly behind is the way to go. Once you get them secured and tidy, put some art, vases, books and whatever you have that is decorative so it doesn't look like nerd central over there.

If you can DIY, I like the idea that asiaone came up with - building a custom tv cabinet and shelf piece that fits over the fireplace brick section. If you paint it the same as the existing wood work, that would probably look best.

I suggest moving all your components next to the TV, and stand them up like books. Then wrap the components with gutted out books, or hide them in decorative boxes just remember to make cut outs for heat ventilations and the IR. Then run all the wires along the back wall and down to outlet.
Or you can relocate all those components inside the cubby hole, mount curtains to hide the cables running up to the tv and cable.

We had a similar issue, but with our kitchen (also a rental). What we did was calculate what we *needed* and *wanted* and what was worth our expenditure. We knew we'd be in the house for at least 2 or 3 years, so we were willing to invest. We spent around $2200 for a carpenter to come in, design something that met our needs, and build and install.

We had some requirements:

Whatever structure we put up would have to be able to be removed fairly easily. We'd need electricity to be accessible. We wanted to be able to hide things (just as you do). The carpenter used almost exclusively IKEA products (tension mounts that were also screwed in for stability) and did an AMAZING job. If you can find someone like that, who can work within a constrained budget and be creative, do that. Because my husband and I aren't carpenters -- we can envision, but not perfectly. We needed someone else to help envision and execute the whole thing.

What I'm thinking is you might want to build out a sort of shelf around the TV on top. It would encase the TV and on either side (or below) you'd store your electronics. You could screw in one or two surge protector thingies to the side or inside, and make those two things the only items whose cords run down to the plug (which is somewhere below, right?) Your carpenter can help secure these single cords in such a way as to minimize the visual clutter. And then a strategically placed floor-vase or sculpture or plant can disguise it further.

I dunno. Just an idea. :)

Like nearly everybody else before has stated -- because you can't drill a hole into the back of the shelving, you should move all your electronics to the top shelf. You don't seem to be using the top shelf for anything else, so why not? Then you can use simple zip ties for a cheap but clean way of organizing the wires. Moving everything to the top of the shelving will also allow you to use the rest of the shelf to decorate or re-organize other items in the room.

Similar to the cordmate, there is an inexpensive cord concealer that comes as a roll. You unroll it and fold it into a box. ~$10, easy to find locally. May be branded differently (mine was in a clear plastic clamshell).

http://www.amazon.com/Belkin-F8B015Q-Hideaway-Concealer-White/dp/B000JJT8GE

If you put stacks of horizontal coffee table sized books on the top shelf and then put one electronic on top of each stack, the cords can get tucked behind the books and then the electronic console looks more like art on top of the books and then if you put more books on the existing shelfs it reads more like a library with kitchy electronics on top.

doh! I spelled shelves wrong ... oh well, decorating is more important than spelling right?

I had a buddy do that with a laptop, he upgraded the hard drive and used an HDMI cable for AV, spent $10 bucks on a few items at Lowe's and it's his Home Theatre PC that is rear TV mounted.

how about this http://www.bluelounge.com/products/cablebox/