Hello Matt,
You probably have version 9.3 or 10 presuming you acquired it within the last 10 years.
An ArcLogistics project includes the myProject.xml file, myProject.mxd file, and the myProject.gdb folder (where "myProject" is the name of your project). These are by default in the "C:\Documents and Settings\All Users\Application Data\ESRI\ArcLogistics\Projects" folder by default (Windows XP), but can also be created anywhere. On Vista/7 they are in the ProgramData area.
Copy the above files/folders for your project(s). Anywhere on the new computer will work; use the default location to avoid needing to browse to open them.
Use a text editor (i.e. notepad) to edit project.xml and repair the paths in there to reflect the street data location on the new computer.
Open the project in ArcLogistics. It should open without errors, but expect the map to be blank and not show any data.
If it does not open, try making a new project on the new computer first (browse to the mxd location, etc.). Doing this will invoke the data licensing behind the scenes. After doing this try opening the copied project again.
Now in ArcLogistics, make a New Project. In the New Project window, pick the project you want to copy as a Template and check the Copy options as needed. The Vehicles, Locations and Specialties will always be copied. After selecting the Template, be sure to pick the Map Template (mxd) again from the MXD folder in the street data location again.
After doing this you should have a new project with the data copied, referencing street data on the local computer, and a map showing street data.
Feel free to call Support Services as well if you want somebody to walk you through this.
Jeff