This car sat up for months, dead battery, I bought at auction, new battery, runs and drives fine... radio does not come on at all. occasionally the control head will light up, volume indicator, but will not indicate any volume level no matter what. Here's the odd thing: If you switch car off and on 6 or 7 times w/out cranking... the control head will come on then, but still no radio or cd will play. You can then drive, switch control off and on continually! I just wonder if whatever is keeping control from coming on ordinarily, is also keeping radio itself from coming on. Somebody must be familiar with the odd things of Volvo radios! Forgot to mention, the radio fuse was missing, I replaced it with hopes! but no... interesting the control head would come on w/fuse missing, made no difference when replaced.

 

We have the same problem with our 2004 XC-90.
My mechanic who services primarily Volvo's, has to reprogram the console unit ( theft protection) with the radio to activate if the unit is replaced. These units are known for failure and buying one yourself without having it reprogrammed will not work.
You might have a unit replaced in your vehicle but not reprogrammed as it sounds just like ours after buying a unit on eBay but to no avail.
Good luck.

BTW, this tranny, if you have the 6 cylinder, are also pronged to fail after about 120k. Keep that tranny ATF changed regularly to hopefully extend its life expectancy