Moscow had plans for "chordal" roads that would connect radial roads in a faster way than concentric-ring roads do. But the land allocated for these roads was all built upon in 1990s, so the traffic problem stays.
As in e.g. NYC, subway in Moscow is generally a faster commute than a car, if there's a station where you need to go. Unfortunately, Moscow subway is also concentric rings + radial lines mostly, and has the same problem of getting somewhere "diagonally" without hitting the crowded central hub stations.
Moscow is still pretty walkable, compared to many US cities, with sidewalks, crossings, etc. Though in areas built past ~1960 you have to be ready to walk for significant distances.
Not exactly what you are looking for and I'm sure there are better examples that I don't know of, but Cologne is made up of concentric rings and at least has one park ring.
