Draft Order as of a few days ago.
http://www.baseballamerica.com/draft/2016-mlb-draft-order/
First Round
1 Phillies
2 Reds
3 Braves
4 Rockies
5 Brewers
6 Athletics
7 Marlins
8 Padres
9 Tigers
10 White Sox
11 Mariners
12 Red Sox
13 Rays
14 Orioles
15 Indians
16 Twins
17 Angels
18 Astros
19 Yankees
20 Rangers
21 Mets
22 Dodgers
23 Blue Jays
24 Royals
25 Pirates
26 Cardinals
27 White Sox (for free agent Jeff Samardzija; Giants gave up first-round pick)
28 Nationals (for free agent Jordan Zimmermann; Tigers gave up second-round pick)
29 Mets (for free agent Daniel Murphy; Nationals gave up first-round pick)
30 Dodgers (for free agent Zack Greinke; Diamondbacks gave up first-round pick)
31 Cardinals (for free agent John Lackey; Cubs gave up first-round pick)
32 Cardinals (for free agent Jason Heyward; Cubs gave up first-round pick)
Competitive Balance Round A
33 Reds
34 Athletics
35 Rockies
36 Dodgers (for not signing Kyle Funkhouser)
37 Diamondbacks
38 Braves (via Marlins)
39 Pirates
Second Round
40 Phillies
41 Reds
42 Braves
43 Rockies
44 Brewers
45 Athletics
46 Marlins
47 Padres
48 White Sox
49 Mariners
50 Red Sox
51 Diamondbacks
52 Rays
53 Orioles
54 Indians
55 Twins
56 Nationals
57 Blue Jays (for not signing Brady Singer)
58 Giants
59 Angels
60 Astros
61 Yankees
62 Rangers
63 Mets
64 Dodgers
65 Blue Jays
66 Royals
67 Pirates
68 Cardinals
69 Orioles (for not signing Jonathan Hughes)
Competitive Balance Round B
70 Padres
71 Indians
72 Twins
73 Brewers
74 Twins (for not signing Kyle Cody)
75 Orioles
76 Rays
Third Round
77 Phillies
78 Reds
79 Braves
80 Rockies
81 Brewers
82 Athletics
83 Marlins
84 Padres
85 Tigers
86 White Sox
87 Mariners
88 Red Sox
89 Diamondbacks
90 Rays
91 Orioles
92 Indians
93 Twins
94 Nationals
95 Giants
96 Angels
97 Astros
98 Yankees
99 Rangers
100 Mets
101 Dodgers
102 Blue Jays
103 Royals
104 Cubs
105 Pirates
106 Cardinals
The 2016 draft order will be in a state of flux over the next few months. There are three ways the draft order can be altered:
1. A team fails to sign a player that it selected in the top three rounds of the previous draft. When this happens, the team is awarded a compensation pick immediately following the pick number of the unsigned player. For example, the Dodgers own the No. 36 pick in 2016 because they failed to sign Kyle Funkhouser, the 35th overall pick in the 2015 draft.
2. A team signs a player who has rejected a qualifying offer from his previous team. When this happens, the signing team forfeits its first unprotected non-compensatory pick, and the players previous team is awarded a pick after the first round. For example, the Diamondbacks forfeited the No. 13 pick in 2016 when they signed Zack Greinke, and Greinkes previous club (the Dodgers) received a pick after the first round.
3. Competitive Balance picks can be traded. For example, the Braves acquired the Marlins Competitive Balance Round A pick in a 13-player deal in July.