The only alternative to having your very own monthly chain letter ("if you don't pass this letter on, you and three other people will lose money!") that I'm aware of is talking with a bank (you'll need to talk with a personal banker) or similar escrow service that will allow you to open an account jointly. The bank may or may not allow you to skip signing individually, but with shared responsibility for the account it may be possible. Effectively this is a limited partnership with a joint bank account, so only one person needs to have signing/withdrawal authority - but this may have tax implications and other complications, too!
What you really want to do is have an effective escrow service: one party accepts a payment on behalf of multiple people, and then that party distributes the money as appropriate to the signed agreements. If - and it's definitely an IF - a company/service would agree to such a thing, you could just sign an agreement once and forward the check to them. They will cash it using their established legal/banking agreements and in accordance with existing regulations, and then distribute the money as agreed.
There will be a fee for this service, of course, and it is subject to this actually being something an escrow company is willing and able to do for you.
Alternatively, if there is only four or five of you and you are all on good terms, then I would spend just a little time setting up a system so this can be quick and relatively painless. Get yourself postage paid envelopes and print out some address labels for everyone, and send these out to everyone.
Have the check sent to person A. A signs the check, puts it into prepaid and pre-addressed envelope and sticks it in mail box (if you are worried about it getting lost you may want to spring for a tracking/certified mailing) to person B. B repeats and sends to C, C sends to D, and D signs and puts in the bank and distributes the funds as appropriate. 5 minutes of time per month and two weeks of calendar waiting (if you don't spring for express shipping), and then you have your money.
In legal matters it really often is better to get permission rather than seek forgiveness, so I would strongly encourage you to talk with a lawyer and/or a banker before doing anything other than the manual mailing method, as you've been planning.
In short, the legal system is inconvenient, grinding, and often deeply antiquated, so don't expect anything terribly convenient or time-saving from it - it's almost as if most of the people involved get paid by the hour, or something, inside a system that seems to practically worship paper and tradition. Go figure!