There are no serious health concerns raised by a cricket infestation. Since these insects rarely appear indoors in groups, any hazard they may present is easily negated by their very small numbers. Most people are simply annoyed by their presence or confuse crickets for cockroaches on first glance.
Several kinds of crickets are found in Arizona. Although they pose no immediate health risks (they do not bite or carry disease) they have been known to eat through everything from wallpaper glue to wool to silk.