Wow, this has generated a lot more controversy than I expected. I thought it was just that I was ignorant about such matters, and that other people would know, but, after reading all the replies so far, I am still confused. The paranoid and the non-paranoid parties both seem to have some good arguments.
To clarify a couple of points that seem to have turned out to be relevant, I and my bank account are in the U.S.A. I do not know for sure where the person/company offering to pay me is based (it does not say on their web site) but I strongly suspect it is outside the U.S. (I rather think I have heard that it is not legal to operate a gambling site within the U.S., although that may not apply to the site in question, as it is a portal site rather than one where games actually happen, or bets are actually placed.) When I dealt with them before, I checked out their site with WHOIS, and it appeared that they were based in Canada. However, all I can get from WHOIS now is that they are registered through GoDaddy.
Perhaps I should fess up about the site’s identity, just in case anyone has any direct knowledge of it. It is pokerlistings . com
The first time I dealt with them, I did originally ask if they could just mail me a check, but they were reluctant, and wanted to deal through PayPal. I did not argue at the time, and I set up a PayPal account (I did not have one before) just for the purpose. Perhaps their reluctance to send a check makes sense in the light of what IamNotSpartacus says in post #37 (and the fact that they are proposing to pay me in U.S. dollars).
It is possible that their delay in getting back to me to renew might indicate that I am no longer really dealing with the same people. However, they were also kind of slow to get back last year too, for their first renewal, but when they did eventually get back in touch, the transaction went through PayPal fine. There are also a couple of other similar sites I have had similar dealings with, and some of them have also tended to be a bit flaky about replying promptly to emails. In one case I did not hear for a few weeks and I took his link down, but then, months later he got back to me and wanted it up again, and his payment got to me OK through PayPal. I guess links on little sites like mine are not a huge priority to these people, even though they are willing to pay for them, but for me that $500 is well worth having. The odd thing this time is that they don’t want to pay through PayPal, especially in the light of what WhyNot says in post #8.
Thank you to everyone who has replied, but it looks like I am just going to have to contact my bank and ask their advice (if I can get through their phone-tree to anyone helpful).