St. Paraskevi Greek Orthodox Shrine Church in Greenlawn, NY: There is a grotto-shrine there, dedicated to St. Paraskevi, with a miracle-working spring of holy water. Many people have been healed of physical infirmities, especially relating to eyes (for which St. Paraskevi is known).
St. Paul Greek Orthodox Cathedral in Hempstead, NY: In addition to amazing gold mosaics, they have a special shrine that contains a Myrrh-Streaming Icon of the Theotokos.
St. Nektarios Greek Orthodox Church in Charlotte, NC: They are blessed with the Holy Relics of St. Nektarios the Wonderworker, for whom they have a shrine. The Church building itself is also an awe-inspiring structure.
Holy Cross Greek Orthodox School of Theology: In addition to the grave of Bishop Gerasimos, an elder to many, there is a relic of the Holy Cross itself, as well as those of several martyrs and saints, including my personal favorite: Saint Panagis (Paisios) of Kefalonia, whose relics are as fragrant as those of St. Gregory Palamas.
Many, many, many monasteries, most of which have Holy Relics. Some that haven't yet been mentioned: St. Anthony's in Arizona; St John Chrysostom in Wisconsin; Holy Dormition in Michigan, etc.