You could always set three cookies, one for each domain. Then just update the cookie as necessary. So when they visit a site (or any page on that site) you update the cookie. The cookie value is nothing more than a Unix Timestamp.
The interstitial page would be if the user is outside the 24-hour time-limit or does not have a cookie. If they don't have cookies enabled, send them to a secondary area, or post information through a hidden form back to the page they came from to start a session on that page. ONly downside is they'd always see the interstitial page if they close their browser.
Basically, it would work something like this:
User A => Cookie Enabled Browser
User B => Cookie Disabled Browser
User A enters site 1. User A has no cookie for Site 1 and is sent to interstitial page. Interstitial page instantiates 3 cookies (one for each domain). User A is redirected to Site 1 with cookies. User A reaches site 1, the cookie is found, the timestamp is within 24 hours, the user is allowed in. Site 1 cookie is updated with new timestamp (if you wish). User A travels to site 2 after 3 hours of viewing site 1. The cookies are found, the site 2 cookie is updated with a new timestamp (if you wish) and the user is allowed in. User A waits 19 hours (for a total of 25 hours) to browse to site 3. Site 3 sees the cookie, note's it's greater than 24 hours, and sends to interstitial page where the process starts over again.
User B enters Site 2. Site 2 sees no cookies and redirects to interstitial page. Interstitial page sees cookies aren't available and posts a hidden form to Site 2 to start a session. Site 2 sees the posted form, starts a session with a timestamp as a value in the array. User browses to Site 1. Site 1 sees no cookie or session, redirects to interstitial page where the process starts over again.
It's overly simplified, but for all intents and purposes you would probably use a mix of both Sessions and Cookies to get what you're after. Not sure if that's exactly what you want, but it's what I would see you wanting.