It sounds from the comment discussion like what you're asking is
Can I have a bunch of sites, each with a subnet under the 10.0.0.0/8 private address space, and communicate between them over leased lines?
The answer is yes, but if you do it over leased lines you're going to need to invest in routers at each location, and configure your edge/default router to use these private-side routers for private (10-net) traffic.
Another option which you semi-alluded to is VPN tunnels -- this may require you to reconfigure (or replace/supplement) your edge routers, but it's probably a better option: Every site has a public internet connection, and you tunnel traffic between them to create a virtual network between the sites.
Cisco has a great, free, PDF eBook that can help you with the VPN design.
The topology discussions are also useful if you decide to go another route (leased lines).
Once you get the VPN or leased lines and routing bits set up the fact that you're working with private addresses isn't material -- your routers treat the traffic like any other internet traffic and figure out where it should go based on (static or BGP-negotiated) routing tables.