Use Cases:

In[65]:= jsonImport = 
 Import["/Users/dreiss/Downloads/bookmarks-2014-09-05.json"];

In[70]:= Cases[jsonImport, z : ("uri" -> _) :> z, [Infinity]]

Out[70]= {"uri" -> "http://community.wolfram.com/dashboard", 
 "uri" -> "http://www.online-convert.com/es/resultado/\n    d7b9f49e6e811d16d321b597ff024cc4", 
 "uri" -> "http://www.spanishdict.com/traductor", 
 "uri" -> "https://www.youtube.com/watch?v=18AIszvxESo", 
 "uri" -> "https://www.youtube.com/watch?v=s-6sMhBH09c", 
 "uri" -> "https://www.youtube.com/watch?v=vFe-R-D9Kig", 
 "uri" -> "https://www.youtube.com/watch?v=6OLOw0EhTBo", 
 "uri" -> "https://www.youtube.com/watch?v=c1gRxQLjjFo", 
 "uri" -> "https://www.youtube.com/watch?v=UjWuOtHGJck", 
 "uri" -> "https://www.youtube.com/watch?v=dDD-x2ncgRk", 
 "uri" -> "https://www.youtube.com/watch?v=ZK0w-vxC2Vo", 
 "uri" -> "https://www.youtube.com/watch?v=B1kL6V-wGy0", 
 "uri" -> "https://www.youtube.com/watch?v=zjUgBGUCMpA", 
 "uri" -> "http://home.comcast.net/~djmpark/index.html#poemlist", 
 "uri" -> "https://www.youtube.com/watch?v=2U_Ln6RUIlU", 
 "uri" -> "https://www.youtube.com/watch?v=4jwMznbBP3M", 
 "uri" -> "https://www.youtube.com/watch?v=As9DnNT3XZA", 
 "uri" -> "https://www.youtube.com/watch?v=xMye-iuQ5Xw", 
 "uri" -> "https://www.youtube.com/watch?v=P0FxZUrIB5M", 
 "uri" -> "place:sort=8&maxResults=10", 
 "uri" -> "http://www.mozilla.com/en-US/firefox/central/", 
 "uri" -> "https://www.youtube.com/watch?v=XmF-eYzka9w", 
 "uri" -> "https://www.youtube.com/watch?v=mFN73-PQX6I", 
 "uri" -> "http://sunsite.univie.ac.at/Mozart/dice/", 
 "uri" -> "https://www.youtube.com/watch?v=_yBxQXombCw", 
 "uri" -> "https://www.youtube.com/watch?v=uH_AZCOhihA"}