Yes you should use your current institution's letterhead. It is important to do so because much of academic hiring turns on prestige and snap judgments. Look professional, look serious, look affiliated with a prestigious place. Use the letterhead.
Edit in response to Nate Eldridge in the comments below:
First, letterhead is to be used in correspondence that constitutes official university business. If helping postdocs and PhD students obtain the positions for which the University has trained them doesn't constitute "official university business," I don't know what would. (Certainly I am instructed to send my undergraduates' letters of recommendation on official university letterhead.)
Second, consider two scenarios. Scenario 1, you apply for a position along with 99 other people. You put the cover letter on letterhead and the other 99 applicants don't. Scenario 2 is flipped, the 99 use letterhead and you don't. In both scenarios, you stick out, which is in general bad. However, they aren't equally bad. Scenario 1 is like showing up to a wedding in a Tux when everyone else is wearing suits. You're a little fancier than you need to be, but nobody's going to think you just weren't putting in care or effort in. Scenario 2, on the other hand, is like showing up to a wedding where everyone is wearing a suit in a shirt and flip flops; you look like you just didn't care what the convention was. So, even if both scenarios were equally likely, I'd still say err on the side of using the letterhead. However, it isn't the case that the scenarios are equally likely--the convention (at least in the US, I'll let others comment about the European or other conventions) is to use letterhead. I'll cite Karen Kelskey, the job market consultant, to corroborate this final point.
I'd hope that no department would allow a faculty who was daft enough to hold the use of letterhead against a candidate anywhere near a search committee.