Bitcoin-friendly Bank As is well known many Bitcoin businesses and in particular exchanges have had a lot of trouble working with banks. Are there any banks that are explicitly Bitcoin-friendly?
Despite the popular consensus on the matter, banks probably aren't specifically Bitcoin-unfriendly.  The repeated bank problems for exchanges probably come down to three main things:


Automatic risk triggers -- Banks have a lot of mechanisms in place that throw up warning bells when a new customer starts moving lots of money around, is rapidly depositing and withdrawing large amounts of money, etc.  In the past, such activities have often been associated with money laundering, crime, and the sorts of things that are generally bad for business.  Even if a Bitcoin business isn't a problem, it looks like one to the risk algorithms.  If, over time, Bitcoin businesses prove to in fact be low-risk while driving significant business for the banks, the algorithms may be adapted to take this into account.  But they're not rewriting their risk algorithms for what is from their perspective one tiny new technology (and that's probably responsible of them).
Regulatory compliance -- When risk triggers are activated, the first thing the bank wants to know is whether a business is fully compliant with anti-money-laundering legislation.  Many Bitcoin businesses and exchanges are, but the fact is that some aren't.  They haven't been prosecuted for it, but they weren't doing all the paperwork that they really should have been.  Sometimes it's not really the business owner's fault either--financial legalities can be complicated and arcane.  Nonetheless, failure to comply is a huge liability for a bank.
Legitimate risks -- Finally, despite their volatility, bitcoins are a much "harder" currency than most bank deposits and transfers, in the sense that a bitcoin transaction can't be reversed, while bank deposits and transfers are all the time.  That means there is a massive incentive for individually shady customers to move ill-gotten gains out of the banking system and into bitcoins, and there's nothing the Bitcoin exchanges and businesses can really do about that.  This is a major risk problem for a bank, who could end up out some serious money if transferred cash is later "charged back."  At least one major bank recently has quoted risk assessments of individual customers, rather than the exchange itself, as the reason for closing an account.


TL;DR:  Bitcoin per se doesn't have as much to do with it as many people think, and in most cases all three of these issues will apply equally at any financial institution regardless of their personal biases, or (more likely) ignorance regarding Bitcoin itself.  So in the sense that you're asking, there may not be such a thing as a Bitcoin-friendly bank.