Of course, a cash offer means the closing can happen faster.
Make sure you provide proof of funds with your offer. While this will allow you to consider removing the appraisal contingency from your offer, I would still recommend you maintain all inspection contingencies and perform all inspections.
A possible offer in your situation, where you're buying with cash, would be for inspections to be completed within 10 days from acceptance, and closing to occur within 21 days from acceptance. Such terms, without a mortgage contingency, may prove to be very enticing to the bank/seller. Do be aware as to whether or not there are multiple offers, which we are seeing on some bank-owned properties. If there are multiple offers, you will have to carefully consider what purchase price to offer.
Your buyer's agent will be able to assist you in determining which terms will make your offer stand out from the rest.