First up will be deliveries to ANA and LOT Polish Airlines. ANA (ZA512, LN 83, JA818A) should deliver tomorrow 5/13/2013 (flyaway the next day) while LOT Polish (ZA272, LN 86, SP-LRC) should deliver the following day on 5/14/2013 (flyaway the next day).
Thomson Airways is expecting their first 787 (ZA317, LN 92, G-TUIA) on 5/30/2013 with the second (ZA318, LN 94, G-TUIB) to follow on 6/2/2013.
JAL can also take delivery this week of ZA184 (LN 89, JA830J) though I haven't heard about a delivery date.
Hainan and China Southern could take their first 787s as early as the end of this month but that could easily slip into June since the delivery is waiting on Chinese aviation authorities certifying the 787 for use by Chinese carriers.
British Airways is looking to take delivery within the next few weeks but it does look like that they'll pick up their first 787s next month.