Getting to Bhubaneswar

There are no direct flights from Dubai to the capital of Odisha, so I'll have to start searching for a flight to a major city with connecting flights.

Searching [flight to Bhubaneswar] , Google offers the following:

Flights to Bhubaneshwar, India (BBI)

No non-stop flights from Dubai From Duration Airlines Bangalore, India 2h 0m Jet Airways Hyderabad, India 1h 30m IndiGo Kolkata, India 1h 15m Jet Airways, IndiGo, Air India Mumbai, India 2h 15m Air India, Jet Airways, IndiGo New Delhi, India 2h 2m Air India, IndiGo Port Blair, India 2h 0m Air India Vishakhapatnam, India 1h 10m IndiGo

To Kolkata
Emirates:   DXB  2:05 AM  =>  CCU  8:15 AM  Nonstop  4h 40m
Jet Konnect:  CCU 10:00 → BBI 11:25 Nonstop 1h 25m
IndiGo:  CCU 12:15  → BBI 13:15  Nonstop  1h 00m

To Mumbai

Air India:  DXB 6:20p → BOM 10:45p Nonstop 2h 55m

Air India:  DXB 11:40p → BOM 04:05p Nonstop 2h 55m

Jet Airways:  Dubai (DXB) 03:20pm  →Mumbai (BOM) 07:55pm Nonstop 3h 05m

Jet Airways:  Dubai (DXB) 06:45pm  →Mumbai (BOM) 11:20pm Nonstop 3h 05m

Jet Airways:  Dubai (DXB) 09:40pm  →Mumbai (BOM) 02:15pm Nonstop 3h 05m

Emirates:  Dubai (DXB) 04:10pm  →Mumbai (BOM) 08:35pm Nonstop 2h 55m

Emirates:  Dubai (DXB) 06:45pm  →Mumbai (BOM) 11:20pm Nonstop 3h 05m

Emirates:  Dubai (DXB) 10:30pm  →Mumbai (BOM) 02:55pm Nonstop 2h 55m

Looks like Mumbai is the way to go!