This post was able the legality of biking directly on Memorial Drive, not about the cycling infrastructure. There are plenty of sources for finding that out, including Google Maps.
Lauren did, in fact, suggest not being in the road. “Memorial Drive is narrow, and cars go way too fast there, so you might not want to ride in the road. But you can if you want to.” This is the perfect (and correct) answer. Cycling on “big roads” such as Memorial Drive is actually much safer than most people think it is—you just have to know how to do it correctly. It takes some training and a bit of nerve at first, but it soon becomes as natural as riding a bike itself. I have personally biked on Memorial Drive, Revere Beach Parkway, and Storrow Drive, always taking the middle of the right lane and changing lanes when needed. While it may be true that “cars go way too fast there”, they always slow down when they approach a slower moving vehicle (like a bicycle) in front of them. If they see me from far enough away (which they will on long straight roads like that), they almost always sort themselves out long before they get to me and pass me with no complaints. As for it being an “unnecessary safety risk”, it’s no more risky than being in a car. And yes, it’s a right and one that some of us are serious about using. We don’t allow motorists to dictate the terms of our rights to us.

Although it might be legal to do it, there are shared bike/walk paths on BOTH sides of Memorial Drive. Wouldn’t you want to emphasize that part in your answer? I would think your website – used as a point of reference for biker’s looking for information – should consider including suggestions for what might also be the SAFEST way. Everyone should be working together here to be safe, considerate and educated. I drive, bike and walk along Memorial Drive and it’s frightening to see bikers try to maneuver onto Memorial Drive when there are usually empty bike/walk baths 4 feet away. It’s an unnecessary safety risk – Sometimes the most important thing isn’t ‘proving your right’.