From Google Maps it seem like it currently is possible to walk all the bridges, although you can't end where you started. Two of the original bridges have gone,

From Google Maps it seem like it currently is possible to walk all the bridges, although you can't end where you started. Two of the original bridges have gone, two more have been built (not in the same locations as the destroyed ones), and there are two additional bridges that are within the city limits now but wouldn't have been in Euler's time.