Grill sounds like an idea, or possibly a large silent fan in the side? There is a gap between the wall due to skirting board, will keep the front looking clean too.
I'd say running anything of a reasonable spec will still get pretty warm quickly.
I had my 2nd gen i7 with 7850 and micro server in a cupboard with 2x 80mm intlet fans and 2x 80mm exhaust fans and the main PC fans ramped up very quickly under the smallest load. Now I've got a 780ti its back on the desk tpp.
Going on you having 3 reasonable size screens I'd say you have a high spec GPU? Personally I'd be keeping the door open during use at the least.