Our TV was bought in 2003 so it's not like it's super new. Our computer is a few years old too. It just so happens they both have s-video capabilities and we just connect the 2 with a s-video cable. In addition we connect sound with a sound cable that connect into the computers headset output and into the TV's left and right sound input.
You really can't see the connection I mention above, they are both connected into the back of the TV and that back of the computer. The wires in the front are for the playstation 2 or the VCR. We could connect the VCR in the back, but would have to unconnected it anytime we wanted to use this set up with the computer. So for us it's better to connect the VCR into the front as needed.
We only have 2 video inputs in the back of the TV and only one has th s-video option. So we use that for the computer and the 2nd on in the back for the DVD. Then the one in the front for either the playstation 2 or the VCR. Whatever we are using at the time. We don't have the computer connect to the TV all the time, it's just easier to leave the wires connected and use them as needed then to try to reach behind there.
Right now we are watching Molly the Barn Owl that way. In the past we have done things like virtual dissection.