I actually was (I think) their first BTC transaction, so here's what I recommend.
http://coinbase.com/ is a REALLY easy way to use BTC. Mind you, all of your transactions are logged, but then this is also true with WU / Moneygram / etc.; unlike WU/Moneygram/etc, you do not have to give your name or address. All anyone sees is the wallet-to-wallet transaction. So if LE were to get a source's moneygram logs, they'd see the transfers to specific real names, whereas with BTC they'd just see the wallet addresses. Unless they're conducting a massive operation to make an example of people, they probably don't have time to track down wallets.
In fact, historically speaking, police have busted BTC deals by using normal routine methods, like looking for someone who revealed too much about themselves and searching their shit. They've never had to actually trace someone's wallet transactions. That's way too time consuming when normal human error is enough to let LE make arrests.
So, if you don't do anything that would get you busted the normal way, you'll be even safer with BTC.