I need to connect to dial-up type modems in several sites around the
world to upload and download data. At the moment, I use a normal
telephone line (dial-up), but that is very expensive (long-distance
phone rates). Is there a way to connect over the internet so that I
don't need to use a telephone line?

Have you ever looked at Satellite Internet? Here are some sites:
www.wildblue.com/
www.direcway.com/

Basically what I need is a dort of VOIP service but that will allow me
to connect to modems (ie data traffic). Sort of Skype-out for data.
That will allow me to connect over the internet and also get very
cheap (local?) calls instead of international rates.

Vonage may be able to help you. However, Vonage expects you to have
'broadband'. If you must use the dialup you mentioned to connect to
the internet at all (instead of just to connect to the sites), then
you can skip this possibility due to the requirement.

 A Vonage modem is made to connect a phone to a router, which is then
connected to the internet. My idea is to try using the dialup modem
you have through the Vonage router. (Connect the phone port of the
Vonage modem to the phone line port of the modem.) If it works, the
receing dialup modems will receive a call from your modem.