In species like this I always try to analyze the variation of the vessels and the amount of axial parenchyma. Even when vessels are diffused, there is a variation in diameter due to the climate where assists in the identification of the real rings.

Growth ring boundaries in Juglans regia wood are commonly marked by a very narrow band of radially flatted and thick-walled fibers. This is extremely easy to see in cross sections prepared for microscopic analysis (http://www.woodanatomy.ch/species.php?code=JGRE#). It's quite easy to get thin cross sections from cores (even with a razor blade), can you go for this technique?

That depend even on what type of equipment are you using. Do you use a microscope, a  tree-ring measurement station (e.g. Lintab 6) or a density multiscanner (e.g Itrax).

Because using a microscope or Lintab it is very difficult to make difference between rings and possible false rings. But using the Itrax it is possible to separate the ring boundries based on density and the concentration of a specific element.

Why these rings would be "false" ? If you can distinguish them, it means that you can observe the band of thick walled fibers ... All these rings are "real" rings because they correspond to real growth slowdown that can happened some years.You can interpret it as a "false-start" of the new growing season or as a temporary growth stop due to others factors. This means that, as said Diego Romeiro, you probably should not observe the decrease of vessels diameter. Moreover, if 4-5 mm is the mean ring thickness, for a very unfavorable season you can expect to observe "real rings" of 1-2 mm, and not of 0,3 mm. But be careful about mean growth ring thickness values, because  they vary during tree ontogeny (i.e. mean ring thickness have a meaning within the same diameter class).