Well, for the record (and anyone else who wants to do their bills online via Linux), here's my experience:
1. First, I took a look at native Linux programs for online bill pay & banking. I had no idea there was such a thing, since I thought that electronic bill pay cost money & therefore precluded FOSS, but a response to a post brought these to my attention. The ones I looked at were:
gnucash, grisbi, kmymoney2, and moneydance
The first three are open source and, as near as I can tell, do not implement on-line bill pay (can someone correct this if it is not true?). I have no idea why they don't implement on-line bill pay. Moneydance is commercial, but does implement on-line bill pay (not through its own site, however, but through my bank, which does supposedly support the protocol moneydance uses, but offers no support for the program itself).
Since I have been using Quicken's bill pay service, this migration requires two steps: I have to start to pay my bills (with Quicken) through my bank, whereupon my bank will get the online payee information (account number, address, etc.). Then, if I start using moneydance, it will tell my bank to pay such and such a bill, which now has the info to do so. Evidently, there's no way to import all the payee information directly into moneydance and have it "initialize" my bank. If someone knows that this capability does exist, I'd appreciate a post.
This is the route I've chosen to initially try, as it gives me a native program on Linux that does all I want to do. The commercial nature of moneydance doesn't bother me so much: $30 for the capability as a one time charge seems fair enough. Quicken requires an annual update, and, although that is free for Quicken bill pay users, Quicken bill pay costs $12.95/month- all in all, moneydance seems a much better deal. However, in order to prep my bank account, it will probably be a month or so before I install it.
2. In looking at wine, I confronted some anomalous information. There is some allusion to not being able to access the internet, specifically for Quicken's autoupdate function, but it does say it downloads banking information, which would obviously require the internet. Supposedly this is due to the lack of support for "schannel", which is some sort of SSL library. There was no specific information as to whether or not it could access the internet to do the bill pay function, but one would certainly think it'd use SSL for that.
I'm sure someone may know more about this than I do, but it was this unknown that finalized my solution to use #1, above.
3. I really didn't look at the virtual machines. I'm sure they work wonderfully, particularly if you have a bunch of windows stuff you need to do on a regular, on-going basis, but taking a brief look at what it takes to install VMWare, I decided it was way beyond what I needed to run only one program and that one only occasionally. When I found out there was a native program, that cinched it. In fact, it was finding out what a great, native system openoffice was that has led me to attempt to clip the M$ umbilical in its entirety.
Note that, since I haven't actually installed and used moneydance yet, I may be revisiting all this in the near future
Thanks to everybody who posted and offered information! Hopefully, this will be of help to somebody.