all your Asian needs can be meet here:
http://www.asiatowncleveland.com/shopping.asp
Asia Market on St Clair and Good Harvest on Payne are my favorites. they are right around the corner from one another. Check them all out. The Asian restaurants and bakery are good too-they are on another link on the website. This area is on the eastern fringes of downtown.
Westside Market
West Side Market, Cleveland Ohio has about everything! Also right around the corner on Lorain ave-just west of the Market by one block is Athens Foods. It is good for all your Greek/Romanian etc food needs. Walk one block more west and you will have a German Market- Import Haus. This area is just west of downtown
My favorite Italian store is Galluchis. It is less than 10 minutes east of downtown.
Italian food and specialty items from Gallucci's.