PILOT AND THREE PASSENGERS DEPARTED BATTLE CREEK MICHIGAN AT 1000, ENROUTE TO BEAVER ISLAND LOCATED IN NORTHERN LAKE MICHIGAN. PILOT WAS RECEIVING VFR ADVISORIES FROM MINNEAPOLIS CENTER, AND REQUESTED A DESCENT NORTH OF TRAVERSE CITY TO GET BELOW A SCATTERED LAYER. DESCENT TOOK THE AIRPLANE BELOW RADAR COVERAGE, AND VFR SERVICES WERE TERMINATED. PILOT ACKNOWLEDGED 1200 TRANSPONDER CODE, AND REPORTED THAT HE WAS VFR AT 1200 FEET MSL. THIS WAS THE LAST CONTACT WITH THE AIRPLANE AT 1150. THE BROTHER OF ONE OF THE PASSENGERS, WHOM THE GROUP WAS GOING TO VISIT, ATTEMPTED TO CALL AND TELL THEM NOT TO COME BECAUSE THE ISLAND WAS ENGULFED BY HEAVY FOG, BUT THEY HAD ALREADY LEFT. HE REPORTED THE AIRPLANE MISSING AT 1700. COORDINATED SEARCH FOUND NO TRACE OF AIRPLANE OR OCCUPANTS. THREE WEEKS LATER, PERSON WALKING ON BEACH NEAR GOOD HART, MICHIGAN, FOUND AN AIRPLANE WHEEL AND WHEEL FAIRING. AIRPLANE AND OCCUPANTS ARE STILL MISSING.