The San Francisco Giants had the most runs by a team in June 2004, with 182 runs.

	TEAM	R	MONTH	G	W%	W	L	AB	H	2B	3B	HR	RBI	BB	HBP	SO	SB	CS	PA	TB	XBH	SH	SF	IBB	GIDP	AVG	OBP	SLG	OPS
1	Giants	182	June 2004	28	.643	18	10	990	282	56	8	34	176	145	16	158	8	6	1,183	456	98	22	10	31	22	.285	.382	.461	.842
2	Phillies	159	June 2004	27	.481	13	14	936	257	49	3	44	152	117	10	189	10	4	1,084	444	96	13	8	9	19	.275	.359	.474	.833
3	Cardinals	157	June 2004	28	.679	19	9	958	271	53	4	27	150	102	6	180	12	6	1,092	413	84	11	15	15	27	.283	.351	.431	.782
4	White Sox	157	June 2004	25	.480	12	13	895	254	63	3	45	152	99	10	189	11	10	1,020	458	111	11	5	6	17	.284	.360	.512	.871
5	Indians	156	June 2004	28	.536	15	13	1,007	285	68	5	29	150	97	11	178	18	15	1,133	450	102	11	7	14	21	.283	.350	.447	.797
6	Rangers	153	June 2004	26	.654	17	9	966	271	45	6	44	144	67	15	187	11	8	1,059	460	95	3	8	5	17	.281	.334	.476	.810
7	Athletics	149	June 2004	27	.630	17	10	944	270	63	3	30	141	90	7	173	10	5	1,054	429	96	2	11	6	21	.286	.349	.454	.803
8	Devil Rays	148	June 2004	26	.769	20	6	907	255	41	12	27	145	103	11	141	20	9	1,037	401	80	6	10	8	16	.281	.358	.442	.800
9	Yankees	140	June 2004	26	.731	19	7	862	221	27	3	46	132	109	15	147	20	6	997	392	76	5	6	11	32	.256	.348	.455	.803
10	Red Sox	133	June 2004	25	.440	11	14	871	244	58	6	35	131	89	9	186	8	3	977	419	99	2	6	5	19	.280	.351	.481	.832
11	Tigers	132	June 2004	26	.538	14	12	916	257	56	9	29	125	81	7	177	17	7	1,019	418	94	8	7	12	13	.281	.341	.456	.798
12	Reds	127	June 2004	26	.462	12	14	911	238	47	0	38	121	90	11	210	16	4	1,028	399	85	10	6	6	18	.261	.333	.438	.771
13	Cubs	126	June 2004	27	.556	15	12	946	256	56	8	33	121	75	6	182	8	8	1,050	427	97	14	9	9	12	.271	.325	.451	.777
14	Orioles	126	June 2004	27	.296	8	19	950	253	47	5	23	118	93	8	149	20	11	1,069	379	75	5	13	5	18	.266	.333	.399	.732
15	Expos	126	June 2004	26	.385	10	16	879	227	47	3	33	120	89	6	149	17	6	1,002	379	83	20	8	7	15	.258	.328	.431	.759
16	Rockies	123	June 2004	26	.346	9	17	890	236	53	4	22	120	82	7	196	12	5	1,003	363	79	16	8	7	23	.265	.329	.408	.737
17	Braves	119	June 2004	27	.444	12	15	917	246	51	5	27	113	87	10	189	11	7	1,034	388	83	13	7	3	23	.268	.336	.423	.759
18	Angels	115	June 2004	27	.407	11	16	952	257	49	8	19	111	77	11	181	31	11	1,062	379	76	16	6	6	22	.270	.330	.398	.728
19	Royals	114	June 2004	27	.444	12	15	919	235	45	7	20	110	81	12	145	15	7	1,028	354	72	9	7	9	19	.256	.322	.385	.707
20	Diamondbacks	113	June 2004	27	.370	10	17	926	249	62	8	19	104	73	3	159	3	5	1,012	384	89	7	3	5	23	.269	.323	.415	.738
21	Blue Jays	112	June 2004	27	.444	12	15	925	239	47	3	22	106	89	11	157	11	4	1,035	358	72	6	4	2	27	.258	.329	.387	.716
22	Twins	111	June 2004	26	.538	14	12	898	221	46	3	33	106	69	13	181	13	4	994	372	82	6	8	1	15	.246	.307	.414	.721
23	Pirates	110	June 2004	28	.321	9	19	935	228	36	5	24	106	54	12	193	7	6	1,024	346	65	17	6	4	12	.244	.292	.370	.662
24	Marlins	110	June 2004	27	.407	11	16	935	237	42	7	24	106	81	8	154	10	7	1,041	365	73	11	6	3	25	.253	.317	.390	.707
25	Dodgers	109	June 2004	26	.462	12	14	875	237	42	3	26	100	79	9	158	21	6	979	363	71	12	4	8	19	.271	.336	.415	.751